Static disorder

Point defects, static disorder, and thermally induced displacements lead to an increase of the background intensity between the spots. Depending on the correlation between the scatters, the background is either homogeneous (no correlation) or... 

In figure 3 and show that the relative thermal motion of the surface atoms is significantly greater than in the bulk metal over the range from 100 - 800 K, This result is expected considering the partial coordination, hence lack of constraint of the surface atoms. A similar result has been found from LEED measurements on a Pt surface. ( ) Significantly, the surface atom disorder when extrapolated to 0 K remains sizable. This static disorder or strain appears to be a result of the interaction of the Ft atoms with the support, a kind of epitaxy to the oxygen (or hydroxyl) surface of the support. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as... 

This can be separated into static disorder and thermal vibrational components ... 

Whereas there is little that one can do to overcome the effects of static disorder, the effects of thermal vibration can be significantly decreased by performing experiments at low temperatures, and, in fact, many solid samples are typically run at liquid nitrogen temperatures just to minimize such effects. An example of the effect of thermal vibration can be ascertained in Fig. 8 A, where the EXAFS amplitude decreases precipitously due to the large vibrational amplitude of the Cu—O bond. In general, failure to consider the effects of thermal vibration and static disorder can result in large... 

In addition to the dynamic disorder caused by temperature-dependent vibration of atoms, protein crystals have static disorder due to the fact that molecules, or parts of molecules, do not occupy exactly the same position or do not have exactly the same orientation in the crystal unit cell. However, unless data are collected at different temperatures, one cannot distinguish between dynamic and static disorder. Because of protein crystal disorder, the diffraction pattern fades away at some diffraction angle 0max. The corresponding lattice distance <7mm is determined by Bragg s law as shown in equation 3.7 ... 

The term exp(-2k2c ) in (6-9) accounts for the disorder of the solid. Static disorder arises if atoms of the same coordination shell have slightly different distances to the central atom. Amorphous solids, for instance, possess large static disorder. Dynamic disorder, on the other hand, is caused by lattice vibrations of the atoms, as explained in Appendix 1. Dynamic disorder becomes much less important at lower temperatures, and it is therefore an important advantage to measure spectra at cryogenic temperatures, especially if a sample consists of highly dispersed particles. The same argument holds in X-ray and electron diffraction, as well as in Mossbauer spectroscopy. 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

On first consideration it may be concluded that if suitable crystals are available X-ray crystallography is the ideal method to decide unambiguously if a candidate compound is, in fact, homoaromatic (Childs et at., 1986a). The bond equalization and planarization associated with homoaromaticity should be readily detected by this means. However, the degree of bond equalization and the size of the homoconjugation gap necessary for homoaromaticity are open to debate (vide infra) (Childs et al., 1986a Haddon, 1988a). In addition, for systems capable of fluxional behaviour, dynamic or static disorder may lead to erroneous conclusions in the interpretation of the X-ray data (see Jackman et al., 1989). In suitable cases very careful X-ray studies can probably avoid this confusion (Dunitz et al., 1988). 

Although it is not known whether this is a dynamic or a static disorder, it is clear from Fig. 2 that the two units result from an inversion at the central carbon, followed by a 180° rotation around the P1-C2 bond. The observation that this carbene has difficulty in maintaining one discrete form in the solid state could well be consistent with the low value for the inversion barrier at carbon, calculated by Hegarty et al.13 (see Section II). 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

Measurements of x ) as a function of temperature should also establish whether or not static disorder dominates the apparent dynamics in a crystalline protein. If the lattice disorder term is the major con-... 

Finally, we note that low-temperature crystallographic studies have been carried out on one nucleic acid, the 5-DNA dodecamer whose room-temperature structure was solved in Dickerson s laboratory (Dickerson, 1981). Refinement at 16 K revealed a large overall drop in B, but some of the atoms in the molecule still had very large B-factors even at this very low temperature. These large residual mean-square displacements were interpreted as demonstrating the presence of static disorder however, by analogy with the results on myoglobin, a disorder which is dynamic at room temperature but becomes frozen into a static distribution at low temperature is also consistent with the observations. It is also possible that the disorder in these atoms is dynamic even at 16 K this point has been considered by Hartmann et al. (1982). 

In practice, the probability distribution often includes static disorders in the crystal. The temperature parameter B in such cases is more properly described as a mean-square displacement parameter. 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

Jnm is the electronic interaction between site m and n in a perfectly ordered lattice and 8Jnm is its variation due to static disorder, and... 

In (1), Hq yields the total energy of system in which the molecules and the lattice are excited, yet there are no interactions between molecules and the lattice. The transfer of an electron from site m to site n is given by //j. Polaronic effects, i.e., effects due to the interaction of the electronic excitation and the lattice, are given by H2 and H. hi H2, the energy of the site is reduced by the interaction with the lattice vibration. In H, the lattice vibration alters the transition probability amplitude from site m to n. The term lattice vibration may refer to inter-molecular or intra-molecular vibrations. Static disorder effects are considered in H4, which describes the changes to the site energy or transition probabihty amplitude by variations in the structure of the molecular sohd. 

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